Theorem 4 (Union theorem, Bell-Dranishnikov, Thm. 25 in [1]}) Let \(X=\bigcup \limits _{\alpha } X_{\alpha }\)  be a metric space where the family \(\lbrace  X_{\alpha } \rbrace  \)  satisfies the inequality \({\rm asdim} \, X_{\alpha } \le n\)  uniformly. Suppose further that for every \(r\)  there is a \(Y_{r}\subset X\)  with \({\rm asdim} \, Y_{r} \le n\)  so that \(d(X_{\alpha } - Y_{r}, X_{\alpha ^{\prime }} - Y_{r} )\ge r\)  whenever \(X_{\alpha } \ne X_{\alpha ^{\prime }}\) . Then \({\rm asdim} \, X \le n\) .

